What is (\partial)/(\partial x)(sqrt(x)e^{-x^2}) ?

The answer to (\partial)/(\partial x)(sqrt(x)e^{-x^2}) is (e^{-x^2})/(2sqrt(x))-2e^{-x^2}xsqrt(x)

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(\partial)/(\partial x)(sqrt(x)e^{-x^2})

\frac{\partial}{\partial x} (x e^{- x^{2}})

\frac{e ^{- x^{2}}}{2 x} - 2 e^{- x^{2}} x x

Solution steps

\frac{\partial}{\partial x} (x e^{- x^{2}})

Apply the Product Rule:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = x, g = e^{- x^{2}}

= \frac{\partial}{\partial x} (x) e^{- x^{2}} + \frac{\partial}{\partial x} (e^{- x^{2}}) x

\frac{\partial}{\partial x} (x) = \frac{1}{2 x}

\frac{\partial}{\partial x} (e^{- x^{2}}) = - 2 e^{- x^{2}} x

= \frac{1}{2 x} e^{- x^{2}} + (- 2 e^{- x^{2}} x) x

Simplify

\frac{1}{2 x} e^{- x^{2}} + (- 2 e^{- x^{2}} x) x : \frac{e ^{- x^{2}}}{2 x} - 2 e^{- x^{2}} x x

= \frac{e ^{- x^{2}}}{2 x} - 2 e^{- x^{2}} x x

y = sec (x) - 2 cos (x), (\frac{π}{3}, 1)

n = 0 \sum \infty (0.7)^{n}

\frac{sec ( y )}{y ^{8}}

\int \frac{1}{e ^{x} ( e ^{x} + 1 )} d x

((x - 2)^{2})^{^{'}}

What is (\partial)/(\partial x)(sqrt(x)e^{-x^2}) ?
The answer to (\partial)/(\partial x)(sqrt(x)e^{-x^2}) is (e^{-x^2})/(2sqrt(x))-2e^{-x^2}xsqrt(x)